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According to the BBC,

Most of the Mexicans who are in the US illegally entered through legal means and then overstayed their visas, not by illicit border-crossing that a wall could prevent.

Looking for data on this, I found a 2006 Pew Report based on INS data from the 90s, an INS report and a GAO report from the early 00s, and there's also a recent DHS report.

But it's not clear to me that there's any recent estimate of either (1) how many Mexicans currently in the U.S. illegally (including those who entered decades ago), entered legally and overstayed, or (2) what percent of the current flow of illegal Mexican immigration to the U.S. is by legal vs. illegal means of entry.

Is there data to support the BBC's claim?

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There are two reasons to believe that less than 17% of illegally-present Mexicans entered using visas but overstayed.

First is the article Review of the Declining Numbers of Visa Overstays in the U.S. from 2000 to 2009 by Robert and John Warren. (Robert Warren was Director of Statistics at the INS, 1986 to 1995). According to Warren:

In Warren (1997), 17% of all unauthorized immigrants from Mexico, El Salvador, Guatemala, and Honduras were estimated to be overstays. For reasons related to the provisions of the Immigration Reform and Control Act of 1986 (a relatively higher proportion of EWI than overstays were legalized), this percent was higher than it would have been in the 2000-2009 decade.

[above "EWI" means "entered without inspection", the opposite of entering legally with a visa]

Secondly, in the GAO report in the OP (tables 6 and 7), the number of Mexicans who were overstays was analyzed by assessing whether or not Mexicans arrested in security operations were overstays or not.

Only 42/472 Mexicans or 9% of Mexicans arrested for being illegally present were found to be overstays.

Additionally, in every single security operation studied, when two or more Mexicans were arrested, it was always true (23 out of 23 operations) that less than half were overstays.

(The comment by @dsollen helps illustrate how BBC would come to the wrong conclusion. It's easy to look at numbers for all illegal immigrants and see a large fraction are due to overstays, and then misapply these statistics to Mexicans. But because Mexico shares a border with the United States, it is much easier for Mexicans to cross illicitly by land without a visa, compare to immigrants from Asia for example. So it is critical to look at statistics that are specific to Mexicans to answer this question.)

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    This doesn't seem definitive - if overstays are significantly less likely to be caught, then they could make up a majority of illegals while still having smaller arrest numbers (not saying that's the case, but I haven't seen numbers either way).
    – Dan Smolinske
    Commented Jan 27, 2017 at 16:20
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    The GAO report itself states " These three alternative data sources on illegal immigrants indicate varying—but uniformly substantial—percentages of overstays: 31 percent, 27 percent, and 57 percent. " It also states numerous times that they believe the estimation of 1/3 of illegals being overstays from previous studies is an under estimation. I don't see how you can use this report as basis for a claim of less then 17% for overstays when the report itself states many times their estimation is for a higher value.
    – dsollen
    Commented Jan 27, 2017 at 18:21
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    @dsollen The question is specifically about Mexican immigrants. Obviously illegal immigrants from New Zealand would be less likely to walk across the border.
    – DavePhD
    Commented Jan 27, 2017 at 18:23
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    @DanSmolinske If you look at the lists of locations (tables 6 and 7) they are very well distributed.
    – DavePhD
    Commented Jan 27, 2017 at 18:27
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    @Sklivvz I disagree because the entire population of illegal workers at each location is arrested in such operations. You could say it is biased towards people who work, or people who work at airports and other high security locations.
    – DavePhD
    Commented Jan 28, 2017 at 17:23

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