I noticed that the speakers with the General American accent occasionally weaken the "h" sound in words like "had" e.g. "You had this and that." becomes kind of like "You ad this and that." (I can't tell whether it's a drop or just a weakening.) Does this phenomenon have a name?
What happens is that in connected, fast speech function words get unaccented and "show reductions of the length of sounds, centralization of vowels toward /ə,ɪ,ʊ/ and the elision of vowels and consonants" (Cruttenden 2014: 273).
As for function words that start with h, the following words get affected: had, has, have, he, her, herself, him, himself, his (Cruttenden 2014: 274). In their weak forms, they may all lose h.
Here's a screenshot of the relevant section from http://www.blackwellpublishing.com/shockey/
This process is not unique to General American. It happens in British dialects as well.
In American English, this is often thought to be a generalization of (de)aspiration as in pit, spit. We typically analyze stop aspiration as addition of aspiration foot-initially, but depending on your theory of how rules can be written, you can state that as deaspirating stops which are not foot-initial. /h/ is also deletable when intervocalic followed by a stressless vowel, i.e. when /h/ is foot-medial (it's always syllable-initial), thus there is a standard alternation in prohíbit, Prò(h)i[ə]bítion. In the nominalization prohibition, where /ɪ/ is not reduced to schwa, /h/ is also not always deleted, though in my dialect the pronunciation with h and ɪ is an unnatural spelling pronunciation.
It's hard to find words that lend themselves to h ~ Ø alternations, so you've found the main source of examples (h-initial pronouns being the other). It also plays a role in people figuring out how to pronounce non-English names based on spelling. The Arabic name Fahim could in principle have stress on either syllable (people vacillate randomly between [hǽmɪd] and [həmíd] for Hamid); but initial stress is out because [fǽhɪm] would put h in foot-medial position, which is a no-no.
However, I have no problem pronouncing [wǽhɪb], [rǽhæt], but it's completely impossible for me to say [átʰa]: the aspirate allophones absolutely cannot be foot-medial. So while h-deletion and aspiration are surely related, they don't literally reduce to one rule. As far as I know, "h-dropping" is applied to context-free deletion of h. I think the Wiki entry is in error in implying that all dialects of English delete h sentence initially, such as "Her mother said 'no'". Of course, you can always define "H-dropping" any way you want, but as far as I know, that term is used for context-free deletion of h.