Strengthening existing 2x4 joists to create a mini attic
Wow, this is quite an effort, but I know how valuable space is in SF. 

I’ll make some assumptions and you let me know if I’m wrong. First, Loads are divided into Live Loads (storage boxes, books, Christmas decorations, etc.) plus Dead Loads (gypsum board ceiling, ceiling joists, 1/2” plywood subfloor, etc.) Generally we add about 3 psf for miscellaneous loads like lights, ductwork, etc. into the dead load. So, here’s your Dead Load:

Gypsum board ceiling: 2.5 psf, plus 2x4’s @ 16” oc: 1.0 psf, plus 1/2” subfloor on 2x4’s: 1.5 psf, plus misc.: 3 psf...for a total Dead Load of 8.0 psf. 

Second, Live Loads are considered uniformly distributed across the storage space...AND loads in the middle third of the span affects the deflection the most AND can cause cracks in the gypsum board ceiling when you exceed the allowable loads. 

Here are your options:

Option 1: Just add subfloor on existing 2x4 @ 16” oc. 

Depending on the species and grade of ceiling joists, 2x4’s @ 16” oc spanning 10’ can support about 12 psf Live Load plus 8 psf Dead Load.

This is certainly the easiest option. I’d suggest you lay out some items you want to store and see if it exceeds 12 psf. (Remember, it’s 12 psf over  the span of several ceiling joists. That is to say, if part of the area is 16 psf and an adjacent area ON THE SAME JOISTS weighs 8 psf...you’re probably ok.)

Option 1A: Same as Option 1, but don’t store anything in the middle third of the span:

With this option, you can store about 20 psf Live Load in the first 1/3 on each side of each span...but nothing in the middle third. 

Option 2: Add 1 - 2x4 between each existing 2x4. (You can sister it to the existing 2x4’s or secure it anywhere between the existing joists.)

This will double the allowable loads discussed in Option 1 and 1A. 

Option 3: Sister 2x6’s to existing 2x4’s. 

2x6’s @ 16” oc spanning 10’ will support about 60 psf  Live Load. If you add that to the existing 2x4’s you can support about 72 psf. However, this is based on the fact that you fasten the 2x6’s to the 2x4’s with adequate nails/screws. (Of course, if they are not fastened together then the allowable Live Load would be 60 psf.)

Option 4: Install 2x4 flat to top of each 2x4 ceiling joist. 

This is not recommended. The extreme fiber in bending on the tension side (on the bottom) is still limited by the lumbers fiber strength for a 2x4 on edge. 

Option 5: Provide supports from ceiling joists up to existing 2x8’s @ 16” on center spanning the same 10’. 

2x8’s @ 16” spanning 10’ can support about 120 psf Live Load. Adding a support (2x4) at mid span will increase the allowable Live Load on the existing 2x4’s up to about 110 psf AND add a point load to the 2x8’s....which could over stress the 2x8’s depending on what the actual load is above the ceiling and what it’s supporting on the second floor. 

If you have a refrigerator or a waterbed above this area I would not do it. Otherwise, the Code requires 40 psf Live Load plus 10 psf for Dead Load for a living space. In this case, I think your 2x8’s would be fine, depending on the species and grade. 

I think your next move is to consider what will be stored above the ceiling and how much it weighs...on average over a 10’ span.